The kinetic energy of a particle of mass m is found by taking one half of the product of the mass and the square of the velocity. Write an expression for the knetic energy of a particle.
KE=.5mv^2
To derive the expression for the kinetic energy of a particle, let's break it down step by step.
Kinetic energy (KE) is defined as the energy possessed by an object due to its motion. The formula for kinetic energy is given as:
KE = 1/2 * m * v^2
Where:
- KE: Kinetic Energy
- m: Mass of the particle
- v: Velocity of the particle
To understand how this expression is derived, let's go through the process:
Step 1: Start with the definition of kinetic energy:
KE = Work
Step 2: Work is the dot product of force (F) and displacement (d):
Work = F * d
Step 3: Force can be described using Newton's second law of motion:
F = m * a
Step 4: Acceleration (a) can be expressed as the rate of change of velocity:
a = (v - u) / t
Where:
- u: Initial velocity of the particle
- t: Time
Step 5: Rearrange the equation from step 4 to find v:
v = u + a * t
Step 6: Substitute the value of a from step 5 into the equation from step 3:
F = m * (v - u) / t
Step 7: Substitute the value of F from step 6 into the equation from step 2:
Work = m * (v - u) / t * d
Step 8: Rearrange the equation from step 7 to solve for t:
t = d / (v - u)
Step 9: Substitute the value of t from step 8 into the equation from step 7:
Work = m * (v - u) / d * d
Step 10: Simplify the equation from step 9:
Work = m * (v - u) / d^2
Step 11: Now, consider the special case where the object starts from rest (u = 0). This simplifies the equation:
Work = m * v / d^2
Step 12: This work done on the object is converted into kinetic energy, so:
KE = Work = m * v / d^2
Step 13: Finally, rearrange the equation from step 12 to isolate kinetic energy:
KE = 1/2 * m * v^2
Hence, the expression for kinetic energy of a particle is:
KE = 1/2 * m * v^2
It's worth noting that this derivation assumes constant acceleration for simplicity. However, the final expression is valid for any velocity, regardless of whether the acceleration is constant or not.